Home » Blog » Evaluate the given limit

Evaluate the given limit

Compute

    \[ \lim_{t \to 0} \frac{\sin (\tan t)}{\sin t}. \]


First, we recall \lim_{x \to 0} \frac{\sin x}{x} = 1. So, we would like to rewrite the given function in a form that looks something like this. So,

    \begin{align*}  \lim_{t \to 0} \frac{\sin (\tan t)}{\sin t} &= \lim_{t \to 0} \left( \frac{1}{\cos t} \cdot \frac{\sin (\tan t)}{\sin t / \cos t} \right)\\  &= \lim_{t \to 0} \left( \frac{1}{\cos t} \cdot \frac{\sin (\tan t)}{\tan t} \right)\\  &= \left(\lim_{t \to 0} \frac{1}{\cos t} \right) \left( \lim_{t \to 0} \frac{\sin (\tan t)}{\tan t} \right)\\  &= 1 \cdot 1 \\  &= 1. \end{align*}

Where we used that \lim_{t \to 0} \tan t = 0 and so \lim_{t \to 0} \frac{\sin (\tan t)}{\tan t} = 1.

One comment

  1. Anonymous says:

    The substitution

    *** QuickLaTeX cannot compile formula:
    y=\tan(t)
    
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:
    http://wordpress.org/extend/plugins/core-control/
    http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

    that you have used to evaluate the limit is not always valid and its conditions should be explicitly mentioned given the nature of the question. In general, if

    *** QuickLaTeX cannot compile formula:
    \lim_{x\to a}g(x)=G
    
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:
    http://wordpress.org/extend/plugins/core-control/
    http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

    , then

    *** QuickLaTeX cannot compile formula:
    \lim_{x\to a}f(g(x))=\lim_{y\to G}f(y)
    
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:
    http://wordpress.org/extend/plugins/core-control/
    http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

    if at least one of the following is true: i)

    *** QuickLaTeX cannot compile formula:
    g(x)\ne G
    
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:
    http://wordpress.org/extend/plugins/core-control/
    http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

    in some punctured neighbourhood of a or ii) f is continuous at G. The first condition is unsatisfied here (\frac{\sin x}{x} is not continuous at 0) but the second one is (

    *** QuickLaTeX cannot compile formula:
    \tan(t)\ne 0
    
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:
    http://wordpress.org/extend/plugins/core-control/
    http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

    in some punctured neighbourhood of 0)

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):